Math Puzzle: Locker Problem

    Like many math problems a good first step is to reduce the problem to a workable size. Let us for this example imagine a scenario with 4 students and 4 lockers. The first student closes each locker, the second student opens the second and fourth lockers, the third student opens the third locker and lastly the fourth student closes the fourth locker. There are a number of patterns we could gather from this example, but I would like to point out that after the nth student the nth locker's state is set and remains unchanged for the remainder of the process. Furthermore, we can observe that the state of each locker is dependent on how many students had interacted with that locker prior and that lockers are interacted with by the students who are their factors. We notice that lockers 2 and 3 are open with 2 factors and lockers 1 and 4 are closed with 1 and 3 factors respectively. From this we can extend a hypothesis that any locker with an even number of factors is closed and any locker with an odd number of factors is open after the process. The set of open lockers describes the set of perfect squares. This is due to the fact that factors come in pairs of 2 and numbers which have a pair of "duplicate" factors will have an odd number of total unique factors.